Group #26104545 — subgroups

Subgroups — #26104545

Order 12032.

636 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243137	C₂	1	2	yes	maximal, complemented
2	#20241209	C₂	1	4	yes	maximal, complemented
3	C₄×#127×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#13093935	—	2	2	no	—
5	#15688878	—	2	2	no	—
6	#15688878	C₂²	1	4	yes	—
7	C₂×#127×C₄₇	C₄	1	1	yes	characteristic
8	C₂×C₄×D₄×C₄₇	—	2	2	no	—
9	C₂×C₄×D₄×C₄₇	C₄	1	2	yes	—
10	C₄₇×#390	—	2	2	no	—
11	C₄₇×#390	C₂²	1	2	yes	—
12	C₄₇×#384	C₄	1	1	yes	characteristic
13	C₄³×C₄₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄₇⋊C₈	—	2	4	no	—
15	#13093935	—	2	8	no	—
16	C₂²×D₄×C₄₇	—	2	2	no	—
17	C₂²×D₄×C₄₇	C₂×C₄	1	2	yes	—
18	C₂³×C₄×C₄₇	—	2	2	no	—
19	#127×C₄₇	—	2	4	no	—
20	C₄×D₄×C₄₇	—	4	8	no	—
21	C₄×D₄×C₄₇	—	4	4	no	—
22	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
23	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
24	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
25	C₂×C₂²⋊C₄×C₄₇	—	2	4	no	—
26	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₄₇	D₄	1	2	yes	—
29	C₂×C₄²×C₄₇	D₄	1	2	yes	—
30	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
31	C₂⁴×C₄₇	—	2	2	no	—
32	C₂×D₄×C₄₇	—	2	8	no	—
33	C₂×D₄×C₄₇	—	2	2	no	—
34	C₂×D₄×C₄₇	—	4	4	no	—
35	C₂×D₄×C₄₇	—	4	2	no	—
36	C₂×D₄×C₄₇	—	4	2	no	—
37	C₂×D₄×C₄₇	—	4	4	no	—
38	C₂×D₄×C₄₇	M₁₆	1	4	yes	—
39	C₂²×C₄×C₄₇	—	2	2	no	—
40	C₂²×C₄×C₄₇	—	4	8	no	—
41	C₂²×C₄×C₄₇	—	4	4	no	—
42	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
47	C₄⋊C₄×C₄₇	—	4	4	no	—
48	C₄⋊C₄×C₄₇	—	4	2	no	—
49	C₂²⋊C₄×C₄₇	—	4	4	no	—
50	C₂²⋊C₄×C₄₇	—	4	8	no	—
51	C₄²×C₄₇	—	2	2	no	—
52	C₄²×C₄₇	—	2	4	no	—
53	C₄²×C₄₇	—	2	2	no	—
54	C₄²×C₄₇	—	2	4	no	—
55	C₄²×C₄₇	—	4	1	no	—
56	C₂×C₄₇⋊C₈	—	2	8	no	—
57	C₂×C₄₇⋊C₈	—	4	4	no	—
58	C₂×C₄₇⋊C₈	—	4	4	no	—
59	C₂³×C₄₇	—	2	4	no	—
60	C₂³×C₄₇	—	4	2	no	—
61	C₂³×C₄₇	—	4	2	no	—
62	C₂³×C₄₇	—	4	4	no	—
63	C₂³×C₄₇	—	4	4	no	—
64	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	D₄×C₄₇	—	4	4	no	—
66	D₄×C₄₇	—	4	4	no	—
67	D₄×C₄₇	—	4	16	no	—
68	C₂×C₄×C₄₇	—	2	2	no	—
69	C₂×C₄×C₄₇	—	2	2	no	—
70	C₂×C₄×C₄₇	—	2	1	no	—
71	C₂×C₄×C₄₇	—	2	1	no	—
72	C₂×C₄×C₄₇	—	2	2	no	—
73	C₂×C₄×C₄₇	—	2	2	no	—
74	C₂×C₄×C₄₇	—	2	2	no	—
75	C₂×C₄×C₄₇	—	2	1	no	—
76	C₂×C₄×C₄₇	—	2	2	no	—
77	C₂×C₄×C₄₇	—	4	2	no	—
78	C₂×C₄×C₄₇	—	4	4	no	—
79	C₂×C₄×C₄₇	—	8	8	no	—
80	C₂×C₄×C₄₇	#100	1	2	yes	—
81	C₂×C₄×C₄₇	#104	1	4	yes	—
82	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
83	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
84	C₄₇⋊C₈	—	4	8	no	—
85	#7637	—	47	1	no	maximal, Sylow
86	C₂²×C₄₇	—	4	4	no	—
87	C₂²×C₄₇	—	4	8	no	—
88	C₂²×C₄₇	—	8	2	no	—
89	C₂²×C₄₇	—	8	4	no	—
90	C₂²×C₄₇	—	8	2	no	—
91	C₂²×C₄₇	#407	1	1	yes	characteristic
92	C₂²×C₄₇	#410	1	1	yes	characteristic
93	C₂²×C₄₇	C₂×#100	1	1	yes	characteristic
94	C₂²×C₄₇	#418	1	2	yes	—
95	C₂²×C₄₇	C₂×#104	1	2	yes	—
96	C₄×C₄₇	—	2	2	no	—
97	C₄×C₄₇	—	2	2	no	—
98	C₄×C₄₇	—	2	4	no	—
99	C₄×C₄₇	—	2	4	no	—
100	C₄×C₄₇	—	4	1	no	—
101	C₄×C₄₇	—	8	2	no	—
102	C₄×#127	D₄₇	1	1	yes	characteristic
103	#1745	—	47	2	no	—
104	#1270	—	47	4	no	—
105	C₂×C₄₇	—	8	4	no	—
106	C₂×C₄₇	#1466	1	1	yes	characteristic
107	C₂×C₄₇	#1490	1	2	yes	—
108	C₂×C₄₇	#1505	1	2	yes	—
109	C₂×C₄₇	#1525	1	1	yes	characteristic
110	C₂×C₄₇	#1555	1	1	yes	characteristic
111	C₂×#127	Dic₄₇	1	1	yes	characteristic
112	C₂×C₄×D₄	—	2	2	no	—
113	C₂×C₄×D₄	Dic₄₇	1	2	yes	—
114	C₂×#97	—	94	2	no	—
115	#390	—	2	2	no	—
116	#390	C₂×D₄₇	1	2	yes	—
117	#384	Dic₄₇	1	1	yes	characteristic
118	C₄³	C₂×D₄₇	1	1	yes	characteristic
119	#336	—	47	4	no	—
120	#336	—	94	2	no	—
121	C₄₇	#7637	1	1	yes	characteristic, complemented, Sylow
122	C₂²×D₄	—	2	2	no	—
123	C₂²×D₄	C₂×Dic₄₇	1	2	yes	—
124	C₂³×C₄	—	2	2	no	—
125	C₂²×C₈	—	94	4	no	—
126	#127	—	2	4	no	—
127	C₄×D₄	—	4	8	no	—
128	C₄×D₄	—	4	4	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
131	C₂×C₂²⋊C₄	—	2	2	no	—
132	C₂×C₂²⋊C₄	—	2	4	no	—
133	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
134	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
135	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
136	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
137	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
138	#97	—	94	8	no	—
139	C₂⁴	—	2	2	no	—
140	C₂×D₄	—	2	8	no	—
141	C₂×D₄	—	2	2	no	—
142	C₂×D₄	—	4	2	no	—
143	C₂×D₄	—	4	2	no	—
144	C₂×D₄	—	4	4	no	—
145	C₂×D₄	—	4	4	no	—
146	C₂×D₄	#10625329	1	4	yes	—
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	—	4	8	no	—
149	C₂²×C₄	—	4	4	no	—
150	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
151	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
152	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
153	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
154	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
155	C₂×C₈	—	94	8	no	—
156	C₂×C₈	—	188	4	no	—
157	C₂×C₈	—	188	4	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	4	no	—
160	C₂²⋊C₄	—	4	4	no	—
161	C₂²⋊C₄	—	4	8	no	—
162	C₄²	—	2	2	no	—
163	C₄²	—	2	4	no	—
164	C₄²	—	2	2	no	—
165	C₄²	—	2	4	no	—
166	C₄²	—	4	1	no	—
167	C₂³	—	2	4	no	—
168	C₂³	—	4	2	no	—
169	C₂³	—	4	2	no	—
170	C₂³	—	4	4	no	—
171	C₂³	—	4	4	no	—
172	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
173	D₄	—	4	4	no	—
174	D₄	—	4	4	no	—
175	D₄	—	4	16	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	—	4	4	no	—
187	C₂×C₄	—	8	8	no	—
188	C₂×C₄	#13093941	1	2	yes	—
189	C₂×C₄	#13093951	1	4	yes	—
190	C₂×C₄	C₂×#10625329	1	2	yes	—
191	C₂×C₄	#13094050	1	4	yes	—
192	C₈	—	188	8	no	—
193	C₂²	—	4	4	no	—
194	C₂²	—	4	8	no	—
195	C₂²	—	8	2	no	—
196	C₂²	—	8	2	no	—
197	C₂²	—	8	4	no	—
198	C₂²	#15689087	1	1	yes	characteristic
199	C₂²	#15689097	1	1	yes	characteristic
200	C₂²	C₂×#13093941	1	1	yes	characteristic
201	C₂²	#15689128	1	2	yes	—
202	C₂²	C₂×#13093951	1	2	yes	—
203	C₄	—	2	2	no	—
204	C₄	—	2	4	no	—
205	C₄	—	2	2	no	—
206	C₄	—	2	4	no	—
207	C₄	—	4	1	no	—
208	C₄	—	8	2	no	—
209	C₂	—	8	4	no	—
210	C₂	#20241729	1	1	yes	characteristic
211	C₂	#20241831	1	2	yes	—
212	C₂	#20241896	1	2	yes	—
213	C₂	#20241980	1	1	yes	characteristic
214	C₂	#20242102	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20243137

C₂

yes

maximal, complemented

#20241209

C₂

yes

maximal, complemented

C₄×#127×C₄₇

C₂

yes

maximal, characteristic

C₂×#13093935

—

#15688878

—

#15688878

C₂²

yes

—

C₂×#127×C₄₇

C₄

yes

characteristic

C₂×C₄×D₄×C₄₇

—

C₂×C₄×D₄×C₄₇

C₄

yes

—

C₄₇×#390

—

C₄₇×#390

C₂²

yes

—

C₄₇×#384

C₄

yes

characteristic

C₄³×C₄₇

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

—

#13093935

—

C₂²×D₄×C₄₇

—

C₂²×D₄×C₄₇

C₂×C₄

yes

—

C₂³×C₄×C₄₇

—

#127×C₄₇

—

C₄×D₄×C₄₇

—

C₄×D₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₂×C₂²⋊C₄×C₄₇

—

C₂×C₂²⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₂⁴×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

M₁₆

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#100

yes

—

C₂×C₄×C₄₇

#104

yes

—

C₂×C₄×C₄₇

C₂×M₁₆

yes

—

C₂×C₄×C₄₇

AGL(1,ℤ/8)

yes

—

C₄₇⋊C₈

—

#7637

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

C₂×#104

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

100

C₄×C₄₇

—

101

C₄×C₄₇

—

102

C₄×#127

D₄₇

yes

characteristic

103

#1745

—

104

#1270

—

105

C₂×C₄₇

—

106

C₂×C₄₇

#1466

yes

characteristic

107

C₂×C₄₇

#1490

yes

—

108

C₂×C₄₇

#1505

yes

—

109

C₂×C₄₇

#1525

yes

characteristic

110

C₂×C₄₇

#1555

yes

characteristic

111

C₂×#127

Dic₄₇

yes

characteristic

112

C₂×C₄×D₄

—

113

C₂×C₄×D₄

Dic₄₇

yes

—

114

C₂×#97

—

115

#390

—

116

#390

C₂×D₄₇

yes

—

117

#384

Dic₄₇

yes

characteristic

118

C₄³

C₂×D₄₇

yes

characteristic

119

#336

—

120

#336

—

121

C₄₇

#7637

yes

characteristic, complemented, Sylow

122

C₂²×D₄

—

123

C₂²×D₄

C₂×Dic₄₇

yes

—

124

C₂³×C₄

—

125

C₂²×C₈

—

126

#127

—

127

C₄×D₄

—

128

C₄×D₄

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

C₂×Dic₄₇

yes

—

131

C₂×C₂²⋊C₄

—

132

C₂×C₂²⋊C₄

—

133

C₂×C₄²

C₂×Dic₄₇

yes

characteristic

134

C₂×C₄²

C₂×Dic₄₇

yes

characteristic

135

C₂×C₄²

C₂²⋊D₄₇

yes

—

136

C₂×C₄²

C₂²⋊D₄₇

yes

—

137

C₂×C₄²

C₂²×D₄₇

yes

characteristic

138

#97

—

139

C₂⁴

—

140

C₂×D₄

—

141

C₂×D₄

—

142

C₂×D₄

—

143

C₂×D₄

—

144

C₂×D₄

—

145

C₂×D₄

—

146

C₂×D₄

#10625329

yes

—

147

C₂²×C₄

—

148

C₂²×C₄

—

149

C₂²×C₄

—

150

C₂²×C₄

C₂²⋊Dic₄₇

yes

characteristic

151

C₂²×C₄

C₂²⋊Dic₄₇

yes

characteristic

152

C₂²×C₄

C₂²⋊Dic₄₇

yes

—

153

C₂²×C₄

C₂²×Dic₄₇

yes

characteristic

154

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

155

C₂×C₈

—

156

C₂×C₈

—

188

—

157

C₂×C₈

—

188

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₂²⋊C₄

—

161

C₂²⋊C₄

—

162

C₄²

—

163

C₄²

—

164

C₄²

—

165

C₄²

—

166

C₄²

—

167

C₂³

—

168

C₂³

—

169

C₂³

—

170

C₂³

—

171

C₂³

—

172

C₂³

C₂×C₂²⋊Dic₄₇

yes

characteristic

173

D₄

—

174

D₄

—

175

D₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

#13093941

yes

—

189

C₂×C₄

#13093951

yes

—

190

C₂×C₄

C₂×#10625329

yes

—

191

C₂×C₄

#13094050

yes

—

192

C₈

—

188

—

193

C₂²

—

194

C₂²

—

195

C₂²

—

196

C₂²

—

197

C₂²

—

198

C₂²

#15689087

yes

characteristic

199

C₂²

#15689097

yes

characteristic

200

C₂²

C₂×#13093941

yes

characteristic

201

C₂²

#15689128

yes

—

202

C₂²

C₂×#13093951

yes

—

203

C₄

—

204

C₄

—

205

C₄

—

206

C₄

—

207

C₄

—

208

C₄

—

209

C₂

—

210

C₂

#20241729

yes

characteristic

211

C₂

#20241831

yes

—

212

C₂

#20241896

yes

—

213

C₂

#20241980

yes

characteristic

214

C₂

#20242102

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₇
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄×#127×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂×#127×C₄₇	C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄×#127	D₄₇
hypocenter	C₄₇	#7637
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#127

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₄₇

commutator

C₂²×C₄×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄×#127×C₄₇

C₂

socle

C₂³×C₄₇

C₂×C₂²⋊C₄

prime radical

C₂×#127×C₄₇

C₄

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#127

N	H ≅ G/N	type
1. #20243137	C₂	semidirect
2. #20241209	C₂	semidirect
121. C₄₇	#7637	semidirect

H ≅ G/N

type

1. #20243137

C₂

semidirect

2. #20241209

C₂

semidirect

121. C₄₇

#7637

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20243137	2
#20241209	2
C₄×#127×C₄₇	2
#7637	47

subgroup

index

#20243137

#20241209

C₄×#127×C₄₇

#7637

Subgroups — #26104545

Conjugacy classes of subgroups

Complements

Maximal subgroups